Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 84

As a result, the periodogram is a biased estimator of the power spectral density.
Nevertheless, if hypothesis 3.1 is verified, which is always the case for a linear
process, then:

{ }

() ()

lim I f = S f [3.22] E

x

N →∞

The periodogram is thus an asymptotically unbiased estimator. Moreover, we
can establish [POR 94] that:

∞

∑

{ }

() () () ( )

mm γ < ∞⇒ I f = S f + ON 1/ E

xx x

m =−∞

Let us now study the asymptotic distribution of the periodogram. This
development is generally done under the hypothesis that the process x ( k ) is linear
[POR 94, PRI 94] or that the cumulants of x ( k ) are absolutely summable [BRI 81].
We will briefly recall, in the second case, the main steps of the demonstration. We
thus move on to hypothesis 3.3.

HYPOTHESIS 3.3. The cumulants are absolutely summable, that is to say:

+∞

∑

( )

∀ k u , … , u < ∞

cum

1 k − 1

uu ,, … =−∞

1 k − 1

Let us emphasize that it is possible to define under this hypothesis the multi-
spectrum of k th order for all values of k , [LAC 97]:

+∞

k = 1

1

∑

= ∑ u e π

− j 2 u f

( ) ( )

Sf , … , f u , … ,

cum

x 1 k − 1 1 k − 1

uu ,, … =−∞

1 k − 1

The first step consists of calculating the asymptotic properties of the discrete
Fourier transform:

N

1

= ∑

() 2

− j π f

()

df xne

N

n = 1

RESULT 3.4. Let k ( N ) be an integer such that () () /

f NkNN f →

for

k k

( ) ,

() () ()

k = 1, …, K when N →∞ . If f N ± f N ≡ 0 mod(l), then df N

k k